Find z (if it exists) satisfying the relations |(z+6)/(z+4 i)|=(5)/(3) and |(z+2)/(z+4)|=1.

Name: Problem 5, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-06T14:49:15-08:00
Duration: 10 min 14 s
Channel: Anas Venkitta
Description: Find z (if it exists) satisfying the relations |(z+6)/(z+4 i)|=(5)/(3) and |(z+2)/(z+4)|=1.

Find z (if it exists) satisfying the relations |(z+6)/(z+4...

Key Concepts

Properties of the Modulus in Fractions

When working with the modulus of a quotient of two complex numbers, it is important to use the property that the modulus of a quotient equals the quotient of the moduli. This property simplifies many complex equations by reducing them to relationships between distances in the complex plane.

Geometric Interpretation of Modulus Equations

Equations that involve moduli often represent geometric loci in the complex plane, such as circles or lines, based on ratios of distances. Recognizing that these equations can be interpreted as conditions on distances helps to convert algebraic equations into geometric problems, making them easier to visualize and solve.

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. They are used to solve equations that do not have real solutions and are represented graphically in the complex plane, with the real part corresponding to the horizontal axis and the imaginary part to the vertical axis.

Modulus of a Complex Number

The modulus (or absolute value) of a complex number measures its distance from the origin in the complex plane. It is computed as the square root of the sum of the squares of the real and imaginary parts. This concept is central when dealing with equations that involve magnitudes, as it transforms complex relationships into real number relationships.

Transcript

00:01 So in the given question we are told to find the complex number is z and the condition given is that z plus 6, z plus 6 divided by z plus 4i, the modulus of this is equal to 5 by 3.

00:25 So this is given in the question and it is also said that z plus 2 divided by z plus 4 is equal to 1.

00:39 So these are the conditions that are given in the question and we are told to find the complex number z.

00:47 So let's start with this, right? so what we can write from this is we can use the property that says z1 divided by z2, the modulus of z1 divided by z 2 can be found as modulus of z1 divided by modulus of z2 and we can take that over here and write it as modulus of z 2, z plus 2 divided by modulus of z plus 4 is equal to 1 from which we can write the modulus of z plus 2 is equal to the modulus of z plus 4.

01:26 Now what this expression geometrically implies is that the distance of the complex number, the distance of the complex number z from the point minus 2 is as the same as is as the same as the same as the distance from the distance from the distance of the complex number of z from minus 4.

02:02 So this is what the expression modulus of z plus 2 is equal to modulus of z plus 4 is giving us, right? so now we can say that the z is lying between a line in between the points minus two zero and minus four zero in the argon plane and such a point would lie on the line we are told that it is equidistant from these points and the point that is at an equal distance from this point is the point minus three zero right minus three zero or we can say that it would come on a line with x coordinate minus 3.

02:51 We can't be sure about the y coordinate but we can be sure about the x coordinate which we can tell is at an equidistant it is equidistant from minus 2 0 and minus 4 0.

03:07 So then we can write that z would definitely be of the form minus 3.

03:15 Plus iy right so we can now say that the x the x value of the complex number z is definitely minus 3 since minus 3 is the point that is equidistant from minus 2 and minus 4 x coordinates in the organ plane so now that we have found a relation of z next what we are going to do is to take the first condition that we were given, that is z plus 6, modulus of z plus 6 divided by modulus of z plus 4i is equal to 5 by 3.

04:02 And from this we can write that 3 times the modulus of z plus 6 is equal to 5 times the modulus of z plus 4i.

04:14 And since we already found that z is of the form minus 3 plus iy, we can substitute for z over here and we will have 3 times modulus of minus 3 plus 6 plus iy is equal to 5 times minus 3 plus iy plus i .y times 4 and we can simplify this as 3 minus 3 is 3 so 3 plus iy is equal to 5 times the modulus of minus 3 plus 4 plus y i right so now on finding the modulus of 3 plus i what we will have is square root of 3 square, square root of 3 square plus y square...

Please give Ace some feedback